Equivariant automorphisms and GIT quotients

Question

Maybe this questions is trivial, so feel free to downvote and I will delete it. The base field for me is $\mathbb{C}$. Let us assume that we have a smooth affine variety $X$ with an action of a reductive group $G$. Let $Y = X//G$ be the good categorical quotient of $X$. I would like to understand equivariant automorphisms of $X$ that preserve the orbits of $G$. If $X$ was a $G$-principal bundle over $Y$ in the classical topology, then such automorphisms would correspond bijectively to sections of the adjoint bundle. However, I do not know what happens in this case.